solve a and b for the following questions.

n$-$butanol $(C_4{H}_{10}O)$ is produced by microbial fermentation of lactose $(C_{12}{H}_{22}{O}_{11})$

using an anaerobic fermenter with a selected strain of $C.$ acetobutylicum.

Conventionally, there are other byproducts, but with the exception of butyric acid $($

$C_4{H}_8{O}_2),$ these are suppressed due to the action of simultaneous microbubble

distillation. The fermentation equation can be written:

$C_{12}{H}_{22}{O}_{11}{C}_4{H}_{10}O+C_4{H}_8{O}_2+4C{O}_2+2H_2$

The fermentation process is set up at $37C$ and atmospheric pressure. Butanol is

removed from the fermenter by a side stream microbubble distiller, which uses an

insoluble carrier gas with initially no $H_2$ nor $C{O}_2$ content.

The density of water is $1000k\frac{g}{m^3}$; n$-$butanol $810k\frac{g}{m^3}$; butyric acid $960k\frac{g}{m^3}$ and

lactose $1520k\frac{g}{m^3}.$ The molecular weight of water is $18\frac{g}{m}ol$; n$-$butanol $74\frac{g}{m}ol$;

butyric acid $88\frac{g}{m}ol$ and lactose $342\frac{g}{m}ol.$

$($a$)$ The best genetically engineered mutant strain of $C.$ acetobutylicum is limited

to a maximum volume fraction of $3\%$ butanol in aqueous media before

completely inhibiting the butanol production. The aqueous fermentation broth

has the density of water, regardless of the lactose concentration and

dissolved gas composition, but varies in density with butanol and butyric acid

concentration. Assuming stochiometric production of all species, what are

the molar concentrations of all species, assuming complete dissolution of

product gases, at the inhibition limit$?$

$($b$)$ A distiller takes the output of the fermenter and removes all the butanol,

butyric acid, hydrogen and carbon dioxide gas from the liquid. A fraction $$ of

the liquid is recycled to the fermenter. The fraction $1-$ is the product

stream. Draw a process diagram for the combined continuous fermentation

and distillation process when the feed stream is an aqueous solution with

volumetric flowrate of $1\frac{L}{h}r$ which has molar flowrates of lactose

$M_{\text{l}\text{a}\text{c}}^F=0\mathrm{.}936\frac{\text{m}\text{o}\text{l}}{hr}$ and water $M_w^F=37\mathrm{.}8\frac{\text{m}\text{o}\text{l}}{hr}.$ The fermenter produces butanol

at the inhibition concentration limit$.$ Draw a process diagram, labelling all

streams. Use molar flow rates for all species. List all your assumptions

necessary to write a material balance.